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With the help of the character analogues of the Euler--Maclaurin summation formula we establish integral representations for the Hardy-Berndt character sums $s_{3,p}\\left( d,c:\\chi\\right) $ and $s_{4,p}\\left( d,c:\\chi\\right) $."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1506.01867","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-06-05T11:18:16Z","cross_cats_sorted":[],"title_canon_sha256":"256780938bce136378a38e11e03c34dbd681f80c54ae92856215371d9fb468cc","abstract_canon_sha256":"040f4837f298c98caf30d0da52f3b663b1c438fc672cdbe15f961590e4b55b43"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:46.719227Z","signature_b64":"R+yVkm6W+Bivug0qkLIXzlJTzVZ9TRVCxvhATdjyEJCRWMldT5UuCJfkT010wo27ZOqPm8/Ch6G2H8AIDNCsCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eae2d2c1f61c8964ede35087f9c957c080f0cf7dc997ee87c1b87c4f2764f80a","last_reissued_at":"2026-05-18T01:09:46.718626Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:46.718626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Character analogues of certain Hardy-Berndt sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"M\\\"um\\\"un Can, Veli Kurt","submitted_at":"2015-06-05T11:18:16Z","abstract_excerpt":"In this paper we consider transformation formulas for \\[ B\\left( z,s:\\chi\\right) =\\sum\\limits_{m=1}^{\\infty}\\sum\\limits_{n=0} ^{\\infty}\\chi(m)\\chi(2n+1)\\left( 2n+1\\right) ^{s-1}e^{\\pi im(2n+1)z/k}. \\] We derive reciprocity theorems for the sums arising in these transformation formulas and investigate certain properties of them. With the help of the character analogues of the Euler--Maclaurin summation formula we establish integral representations for the Hardy-Berndt character sums $s_{3,p}\\left( d,c:\\chi\\right) $ and $s_{4,p}\\left( d,c:\\chi\\right) $."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.01867","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1506.01867","created_at":"2026-05-18T01:09:46.718729+00:00"},{"alias_kind":"arxiv_version","alias_value":"1506.01867v1","created_at":"2026-05-18T01:09:46.718729+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.01867","created_at":"2026-05-18T01:09:46.718729+00:00"},{"alias_kind":"pith_short_12","alias_value":"5LRNFQPWDSEW","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"5LRNFQPWDSEWJ3PD","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"5LRNFQPW","created_at":"2026-05-18T12:29:05.191682+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5LRNFQPWDSEWJ3PDKCD7TSKXYC","json":"https://pith.science/pith/5LRNFQPWDSEWJ3PDKCD7TSKXYC.json","graph_json":"https://pith.science/api/pith-number/5LRNFQPWDSEWJ3PDKCD7TSKXYC/graph.json","events_json":"https://pith.science/api/pith-number/5LRNFQPWDSEWJ3PDKCD7TSKXYC/events.json","paper":"https://pith.science/paper/5LRNFQPW"},"agent_actions":{"view_html":"https://pith.science/pith/5LRNFQPWDSEWJ3PDKCD7TSKXYC","download_json":"https://pith.science/pith/5LRNFQPWDSEWJ3PDKCD7TSKXYC.json","view_paper":"https://pith.science/paper/5LRNFQPW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1506.01867&json=true","fetch_graph":"https://pith.science/api/pith-number/5LRNFQPWDSEWJ3PDKCD7TSKXYC/graph.json","fetch_events":"https://pith.science/api/pith-number/5LRNFQPWDSEWJ3PDKCD7TSKXYC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5LRNFQPWDSEWJ3PDKCD7TSKXYC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5LRNFQPWDSEWJ3PDKCD7TSKXYC/action/storage_attestation","attest_author":"https://pith.science/pith/5LRNFQPWDSEWJ3PDKCD7TSKXYC/action/author_attestation","sign_citation":"https://pith.science/pith/5LRNFQPWDSEWJ3PDKCD7TSKXYC/action/citation_signature","submit_replication":"https://pith.science/pith/5LRNFQPWDSEWJ3PDKCD7TSKXYC/action/replication_record"}},"created_at":"2026-05-18T01:09:46.718729+00:00","updated_at":"2026-05-18T01:09:46.718729+00:00"}